Self-sustained oscillators frequently appear as crucial elements in the electrical devices that surround us in our highly technological environment. Essentially, oscillators are devices generating a periodic signal at an inherent frequency, so their primary function is to provide a time reference (clock) or frequency reference. An ideal self-sustained oscillator is mathematically described as a limit cycle in the phase space of dynamical variables, or equivalently as a periodic solution of a set of autonomous differential equations. The ideal oscillator can be described in terms of a steadily increasing phase variable corresponding to the phase space point advancing around the limit cycle, with a 2π-phase change corresponding to a period of the motion. This phase is highly sensitive to additional stochastic terms (noise) in the equations of motion, as the appearance of periodicity without an external time reference implies the freedom to drift in the phase direction. In the power spectrum of the oscillator output the stochastic phase dynamics lead to a broadening of the spectral peaks, that are perfectly discrete in the ideal case [see M. Lax, Phys. Rev., vol. 160, pp. 290-307, 1967; A. Demir et al., “Phase Noise in Oscillators: a Unifying Theory and Numerical Methods for Characterization,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 47, no. 5, pp. 655-674, 2000], and a degradation of the performance as a time or frequency reference. Thus, an essential task in the design of a good oscillator is to reduce the effects of the noise present in the system on the oscillator phase. See, for example, Greywall et al., “Evading Amplifier Noise in Nonlinear Oscillators,” Phys. Rev. Lett., 72, 19, 2992-2995, (1994).
Oscillator phase noise is described in, for example, US Patent Publication Nos. 2005/0046505 and 2012/0299651.